Results 1 to 5 of 5

Math Help - Analysis Questions

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    154

    Analysis Questions

    1. Prove that there is no largest prime.

    Proof: Assume by contradiction that there is a largest prime  n . Then its divisors are  1 and  n . Then how would I continue?


    2. If  n is a positive integer, prove that the algebraic identity  a^{n} - b^{n} = (a-b) \sum_{0}^{n-1} a^{k} b^{n-1-k} . So use induction on  n ?

    3. If  2^{n}-1 is prime, prove that  n is prime. Then  2^{n} is not prime. But this does not imply that  n is prime.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Feb 2008
    Posts
    14
    for your first question: first imagine there is a largest prime number, X.

    Think about how you could prove there is a larger one.

    hint: think about factorials
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,803
    Thanks
    114
    Quote Originally Posted by heathrowjohnny View Post
    1. Prove that there is no largest prime.

    Proof: Assume by contradiction that there is a largest prime  n . Then its divisors are  1 and  n . Then how would I continue?


    ...
    Have a look here: http://www.mathhelpforum.com/math-he...t+prime+number
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by heathrowjohnny View Post
    3. If  2^{n}-1 is prime, prove that  n is prime. Then  2^{n} is not prime. But this does not imply that  n is prime.
    Yes. If n=pq where p,q>1 then 2^n - 1 = (2^q)^p - 1 and use the factorization trick above. Which means it cannot be prime.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Quote Originally Posted by earboth View Post
    Good link.

    OP - You're on the right track with assuming that there exists a largest prime and heading towards a contradiction. Call the largest prime p_n. Take the product p_{1}p_{2}...p_n. This is obviously not prime. What happens if you add 1 to the product? What divides p_{1}p_{2}...p_n +1?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. analysis questions
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: November 28th 2009, 04:07 AM
  2. analysis questions ..
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 28th 2009, 06:45 AM
  3. A few analysis questions
    Posted in the Calculus Forum
    Replies: 8
    Last Post: January 9th 2009, 10:46 PM
  4. 2 complex analysis questions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 11th 2008, 01:22 PM
  5. analysis questions
    Posted in the Calculus Forum
    Replies: 15
    Last Post: October 15th 2006, 11:18 PM

Search Tags


/mathhelpforum @mathhelpforum